Question

given $m \parallel n$, find the value of $x$ and $y$.
(there is a diagram with two parallel lines $m$ and $n$, and a transversal. on line $m$, an angle is $(3x + 8)^circ$. on line $n$, two angles are $(6x + 19)^circ$ and $(2y + 1)^circ$.)
answer
$x = \square$ $y = \square$

Explanation:

Step1: Identify supplementary angles

Since \( m \parallel n \), the consecutive interior angles \( (6x + 19)^\circ \) and \( (3x + 8)^\circ \) are supplementary (they add up to \( 180^\circ \)). So we set up the equation:
\( (6x + 19) + (3x + 8) = 180 \)

Step2: Solve for \( x \)

Combine like terms:
\( 6x + 3x + 19 + 8 = 180 \)
\( 9x + 27 = 180 \)
Subtract 27 from both sides:
\( 9x = 180 - 27 \)
\( 9x = 153 \)
Divide both sides by 9:
\( x = \frac{153}{9} = 17 \)

Step3: Identify vertical angles

The angle \( (3x + 8)^\circ \) and \( (2y + 1)^\circ \) are vertical angles (equal), so we set up the equation:
\( 3x + 8 = 2y + 1 \)

Step4: Substitute \( x = 17 \) and solve for \( y \)

Substitute \( x = 17 \) into the equation:
\( 3(17) + 8 = 2y + 1 \)
\( 51 + 8 = 2y + 1 \)
\( 59 = 2y + 1 \)
Subtract 1 from both sides:
\( 58 = 2y \)
Divide both sides by 2:
\( y = \frac{58}{2} = 29 \)

Answer:

\( x = 17 \), \( y = 29 \)